Theoretical & Industrial Design of Aerofoils P M V Subbarao Professor Mechanical Engineering Department An Objective Invention ……
Transformation for Inventing a Machine A large amount of airfoil theory has been developed by distorting flow around a cylinder to flow around an airfoil. The essential feature of the distortion is that the potential flow being distorted ends up also as potential flow. The most common Conformal transformation is the Jowkowski transformation which is given by To see how this transformation changes flow pattern in the z (or x - y) plane, substitute z = x + iy into above expression.
This gives For a circle of radius r in z plane is transformed in to an ellipse in - planes:
Flow past circular cylinder in Z-plane is seen as flow past an elliptical cylinder of c=0.8 in – plane.
Flow past circular cylinder in Z-plane is seen as flow past an elliptical cylinder of c=0.9 in – plane.
Flow past circular cylinder in Z-plane is seen as flow past an elliptical cylinder of c= 1.0 in – plane.
Translation Transformations If the circle is centered in (0, 0) and the circle maps into the segment between and lying on the x axis; If the circle is centered in (x c,0), the circle maps in an airfoil that is symmetric with respect to the x axis; If the circle is centered in (0,y c ), the circle maps into a curved segment; If the circle is centered in and (x c, y c ), the circle maps in an asymmetric airfoil.
Final Remarks One of the troubles with conformal mapping methods is that parameters such as x c and y c are not so easily related to the airfoil shape. Thus, if we want to analyze a particular airfoil, we must iteratively find values that produce the desired section. A technique for doing this was developed by Theodorsen. Another technique involves superposition of fundamental solutions of the governing differential equation. This method is called thin airfoil theory.
Airfoil Design Methods The process of airfoil design proceeds from a knowledge of the boundary layer properties and the relation between geometry and pressure distribution. The goal of an airfoil design varies. Some airfoils are designed to produce low drag (and may not be required to generate lift at all.) Some sections may need to produce low drag while producing a given amount of lift. In some cases, the drag doesn't really matter -- it is maximum lift that is important.
More Complex Goals for Design of Aerofoil The section may be required to achieve this performance with a constraint on thickness, or pitching moment, or off-design performance, or other unusual constraints. One approach to airfoil design is to use an airfoil that was already designed by someone. This "design by authority" works well when the goals of a particular design problem happen to coincide with the goals of the original airfoil design. Methods of Design: Direct & Inverse Methods
Direct Methods for Airfoil Design The direct airfoil design methods involve the specification of a section geometry and the calculation of pressures and performance. One evaluates the given shape and then modifies the shape to improve the performance. The two main sub problems in this type of method are ; the identification of the measure of performance the approach to changing the shape so that the performance is improved The simplest form of direct airfoil design involves starting with an assumed airfoil shape (such as a NACA airfoil), determining the characteristic of this section that is most problem some, and fixing this problem.
This process of fixing the most obvious problems with a given airfoil is repeated until there is no major problem with the section. The design of such airfoils, does not require a specific definition of a scalar objective function, but it does require some expertise to identify the potential problems and often considerable expertise to fix them. Let's look at a simple (but real life!) example.
Case Study A company is in the business of building turbines for aero engines. They decide to use a version of one of Bob Liebeck's very high lift airfoils. The pressure distribution at a lift coefficient of 1.4 is shown below. Note that only a small amount of trailing edge separation is predicted. Actually, the airfoil works quite well, achieving a C lmax of almost 1.9 at a Reynolds number of one million.
Real Performance At lower angles of attack, the turbine seemed to drop out the load. The plot above the pressure distribution with a Cl of 0.6.
Complex Design Objectives Sometimes the objective of airfoil design can be stated more positively than, "fix the worst things". To reduce the drag at high speeds while trying to keep the maximum C l greater than a certain value. This could involve slowly increasing the amount of laminar flow at low C l 's and checking to see the effect on the maximum lift. Minimize C d with a constraint on C lmax. Maximize L/D or C l 1.5 /C d or C lmax / C d @ C ldesign. The selection of the figure of merit for airfoil sections is quite important and generally cannot be done without considering the rest of the cascade. In order to build a cascade with maximum L/D, one may not build a section with maximum L/D because the section C l for best C l /C d is different from the cascade C l for best C l /C d.
Inverse Design Another type of objective function is the target pressure distribution. It is sometimes possible to specify a desired Cp distribution and use the least squares difference between the actual and target Cp's as the objective. This is the basic idea behind a variety of methods for inverse design. Airfoil theory can be used to solve for the shape of the camber line that produces a specified pressure difference on an airfoil in potential flow. The second part of the design problem starts when one has somehow defined an objective for the airfoil design. This stage of the design involves changing the airfoil shape to improve the performance. This may be done in several ways: 1. By hand, using knowledge of the effects of geometry changes on Cp and Cp changes on performance. 2. By numerical optimization, using shape functions to represent the airfoil geometry and letting the computer decide on the sequence of modifications needed to improve the design.